Mathematics

=== Discrete mathematics ===
{{Main|Discrete mathematics}}
[[File:Markovkate_01.svg|right|thumb|A diagram representing a two-state [[Markov chain]]. The states are represented by 'A' and 'E'. The numbers are the probability of flipping the state.]]
Discrete mathematics, broadly speaking, is the study of individual, [[Countable set|countable]] mathematical objects. An example is the set of all integers.<ref>{{cite journal |last=Franklin |first=James |author-link=James Franklin (philosopher) |date=July 2017 |title=Discrete and Continuous: A Fundamental Dichotomy in Mathematics |journal=Journal of Humanistic Mathematics |volume=7 |issue=2 |pages=355–378 |url=https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1334&context=jhm |doi=10.5642/jhummath.201702.18 |doi-access=free |issn=2159-8118 |lccn=2011202231 |oclc=700943261 |s2cid=6945363 |access-date=February 9, 2024}}</ref> Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply.{{efn|However, some advanced methods of analysis are sometimes used; for example, methods of [[complex analysis]] applied to [[generating series]].}} [[Algorithm]]s{{emdash}}especially their [[implementation]] and [[computational complexity]]{{emdash}}play a major role in discrete mathematics.<ref>{{cite book |last=Maurer |first=Stephen B. |editor1-last=Rosenstein |editor1-first=Joseph G. |editor2-last=Franzblau |editor2-first=Deborah S. |editor3-last=Roberts |editor3-first=Fred S. |editor3-link=Fred S. Roberts |year=1997 |chapter=What is Discrete Mathematics? The Many Answers |pages=121–124 |title=Discrete Mathematics in the Schools |series=DIMACS: Series in Discrete Mathematics and Theoretical Computer Science |volume=36 |publisher=[[American Mathematical Society]] |doi=10.1090/dimacs/036/13 |isbn=0-8218-0448-0 |issn=1052-1798 |lccn=97023277 |oclc=37141146 |s2cid=67358543 |chapter-url={{GBurl|id=EvuQdO3h-DQC|p=121}} |access-date=February 9, 2024}}</ref>

The [[four color theorem]] and [[Kepler conjecture|optimal sphere packing]] were two major problems of discrete mathematics solved in the second half of the 20th century.<ref>{{cite book |last=Hales |first=Thomas C. |title=Turing's Legacy |author-link=Thomas Callister Hales |editor-last=Downey |editor-first=Rod |editor-link=Rod Downey |year=2014 |pages=260–261 |chapter=Turing's Legacy: Developments from Turing's Ideas in Logic |publisher=[[Cambridge University Press]] |series=Lecture Notes in Logic |volume=42 |doi=10.1017/CBO9781107338579.001 |isbn=978-1-107-04348-0 |lccn=2014000240 |oclc=867717052 |s2cid=19315498 |chapter-url={{GBurl|id=fYgaBQAAQBAJ|p=260}} |access-date=February 9, 2024}}</ref> The [[P versus NP problem]], which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of [[Computationally expensive|computationally difficult]] problems.<ref>{{cite conference |last=Sipser |first=Michael |author-link=Michael Sipser |date=July 1992 |title=The History and Status of the P versus NP Question |conference=STOC '92: Proceedings of the twenty-fourth annual ACM symposium on Theory of Computing |pages=603–618 |doi=10.1145/129712.129771 |s2cid=11678884}}</ref>

Discrete mathematics includes:<ref name=MSC/><!-- Scope of [[Discrete Mathematics (journal)]] [https://www.journals.elsevier.com/discrete-mathematics]The research areas covered by Discrete Mathematics include graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, discrete probability, and parts of cryptography.

Discrete Mathematics generally does not include research on dynamical systems, differential equations, or discrete Laplacian operators within its scope. It also does not publish articles that are principally focused on linear algebra, abstract algebraic structures, or fuzzy sets unless they are highly related to one of the main areas of interest. Also, papers focused primarily on applied problems or experimental results fall outside our scope.

In [[Discrete Mathematics and Computer Science]] [https://dmtcs.episciences.org/page/policies]
General

Analysis of algorithms

Automata, logics and semantics

Combinatorics

Discrete algorithms

Distributed Computing and networking

Graph Theory
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* [[Combinatorics]], the art of enumerating mathematical objects that satisfy some given constraints. Originally, these objects were elements or [[subset]]s of a given [[set (mathematics)|set]]; this has been extended to various objects, which establishes a strong link between combinatorics and other parts of discrete mathematics. For example, discrete geometry includes counting configurations of [[geometric shape]]s
* [[Graph theory]] and [[hypergraph]]s
* [[Coding theory]], including [[error correcting code]]s and a part of [[cryptography]]
* [[Matroid]] theory
* [[Discrete geometry]]
* [[Discrete probability distribution]]s
* [[Game theory]] (although [[continuous game]]s are also studied, most common games, such as [[chess]] and [[poker]] are discrete)
* [[Discrete optimization]], including [[combinatorial optimization]], [[integer programming]], [[constraint programming]]
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